Question: In the right triangle shown, $\angle A = 30^\circ$ and $AB = 8\sqrt{3}$. How long is $BC$ ? $A$ $C$ $B$ $x$ $8\sqrt{3}$
Solution: We know the length of the hypotenuse of this triangle, and want to find the length of a leg. What mathematical relationship is there between a right triangle's legs and its hypotenuse? We can use either sine (opposite leg over hypotenuse) or cosine (adjacent leg over hypotenuse). This is a 30-60-90 triangle, so we know what the values of sine and cosine are at each angle of the triangle. Let's try using cosine: $A$ $C$ $B$ $x$ $8\sqrt{3}$ ${60}^{\circ}$ Cosine is adjacent over hypotenuse (SOH CAH TOA), so $\cos {60}^{\circ} = \dfrac{x}{8\sqrt{3}}$ . We also know that $\cos{60}^{\circ} = \dfrac{1}{2}$ Solving for $x$ , we get $ x = 8\sqrt{3} \cdot \cos{60}^{\circ}$ $ x = 8\sqrt{3} \cdot \dfrac{1}{2}$ So, $x = 4\sqrt{3}$.